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Chapter 5: Problem 59

\(\left(p^{3}-3 p^{2} q\right)+\left(2 p q^{2}+4 q^{3}\right)-\left(3 p^{2} q+p q^{2}\right)\)

Short Answer

Expert verified
\[ p^{3} - 6p^{2}q + pq^{2} + 4q^{3} \]

Step by step solution

01

- Write the Expression

The given expression is: \[ \big(p^{3}-3 p^{2} q\big)+\big(2 p q^{2}+4 q^{3}\big)-\big(3 p^{2} q+p q^{2}\big) \]
02

- Distribute the Negative Sign

Distribute the negative sign across the third group of terms: \[ \big(p^{3}-3 p^{2} q\big) + \big(2 p q^{2} + 4 q^{3}\big) - 3 p^{2} q - p q^{2} \]
03

- Combine Like Terms

Combine like terms in the expression: \[ p^{3} -3p^{2}q -3p^{2}q + 2pq^{2} - pq^{2} + 4q^{3} \] Combine further: \[ p^{3} - 6p^{2}q + pq^{2} + 4q^{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
Combining like terms is an essential skill in simplifying polynomial expressions.
Like terms are terms that have the same variable and the same exponent.
When you combine like terms, you add or subtract their coefficients, which are the numerical parts of the terms.
In our given exercise, we see several terms that can be combined:
  • For example, \(-3p^{2}q\) and \(-3p^{2}q\) can be combined because both terms contain \(p^{2}q\)
Combine these to get \(-6p^{2}q\).
  • Also, \(2pq^{2}\) and \(-pq^{2}\) are like terms because they both contain \(pq^{2}\).
Combining these gives us \(pq^{2}\).
Always ensure to carefully match the variables and their exponents when combining like terms!
Distributing Negative Sign
Distributing a negative sign is another crucial concept in simplifying expressions. When you distribute a negative sign across a group of terms, you are effectively multiplying each term within the parentheses by \(-1\).
In the given exercise, we distribute the negative sign over the third group of terms:
From \(-\big(3 p^{2} q + p q^{2}\big)\) to \(-3 p^{2} q - p q^{2}\).
This changes the signs of all the terms within the parentheses.
It's essential to handle each term separately to avoid errors. Ensure each term inside the parentheses changes its sign when distributing the negative sign.
Simplifying Expressions
Simplifying expressions is the process of combining like terms and distributing signs to make an expression easier to understand.
For our given problem, we first distributed the negative sign and then combined like terms. The original expression was:
\(\big(p^{3}-3 p^{2} q\big) + \big(2 p q^{2}+4 q^{3}\big) - \big(3 p^{2} q+p q^{2}\big)\).
  • After distributing the negative sign, it becomes:
\(\big(p^{3} - 3 p^{2} q\big) + \big(2 p q^{2} + 4 q^{3}\big) - 3 p^{2} q - p q^{2}\).
  • Combine like terms to get:
\(\big(p^{3} - 6 p^{2} q + p q^{2} + 4 q^{3}\big)\).
This is the simplified form of our expression.
Always remember to follow the proper steps of simplifying by ensuring each term is handled accurately and that all like terms are correctly combined.

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